We consider the numerical solution of diffusion problems in for and for in dimension . We use a wavelet based sparse grid space discretization with mesh-width and order , and discontinuous Galerkin time-discretization of order on a geometric sequence of many time steps. The linear systems in each time step are solved iteratively by GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an -error of for where is the total number of operations, provided that the initial data satisfies with and that is smooth in for . Numerical experiments in dimension up to confirm the theory.
@article{M2AN_2004__38_1_93_0, author = {Petersdorff, Tobias Von and Schwab, Christoph}, title = {Numerical solution of parabolic equations in high dimensions}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, volume = {38}, year = {2004}, pages = {93-127}, doi = {10.1051/m2an:2004005}, mrnumber = {2073932}, zbl = {1083.65095}, language = {en}, url = {http://dml.mathdoc.fr/item/M2AN_2004__38_1_93_0} }
Petersdorff, Tobias Von; Schwab, Christoph. Numerical solution of parabolic equations in high dimensions. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) pp. 93-127. doi : 10.1051/m2an:2004005. http://gdmltest.u-ga.fr/item/M2AN_2004__38_1_93_0/
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